# Statistics | Revision Notes

# Statistics

- The mean x̅ of n values x
_{1}, x_{2}, x_{3}, ...... x_{n}is given by

**Mean of grouped data (without class-intervals)****Direct method :**If the frequencies of n observations x_{1}, x_{2}, x_{3}, ..... x_{n}be f_{1}, f_{2}, f_{3}, ..... f_{n}respectively, then the mean x̅ is given by

**Deviation method**or**Assumed mean method**

In this case, the mean x is given by

Where,*a*= assumed mean, Σ*f*= total frequency,_{i}*d*=_{i}*x*_{i}– a

Σ*f*(_{i }*x*) = sum of the products of deviations and corresponding frequencies._{i }– a

**Mean of grouped data (with class-intervals)**

In this case the class marks are treated as*x*._{i}

**Direct method**

If the frequencies corresponding to the class marks x_{1}, x_{2}, x_{3}, ........ x_{n}be f_{1}, f_{2}, f_{3}, ........ f_{n}respectively, then mean x̅ is given by

**Deviation or Assumed mean method**

In this case the mean x is given by

Where, a = assumed mean, Σ*f*= total frequency and_{i}*d*_{i}= x_{i }– a**Step Deviation method**

In this case we use the following formula.

Where, a = assumed mean, Σ*f*= total frequency, h = class-size, ._{i}

**Mode**is that value among the observations which occurs most often i.e., the value of the observation having the maximum frequency.- If in a data more than one value have the same maximum frequency, then the data is said to be
**multimodal**. - In a grouped frequency distribution, the class which has the maximum frequency is called the
**modal class**. - We use the following formula to find the mode of a grouped frequency distribution.

- where

*l*= lower limit of modal class,

*h*= size of the class-interval,

*f*= frequency of the modal class,_{1}

*f*= frequency of the class preceding the modal class,_{0}

*f*= frequency of the class succeeding the modal class._{2}

- where
**Median**is the value of the middle most item when the data are arranged in ascending or descending order of magnitude.**Median of ungrouped data**- If the number of items n in the data is
**odd**, then

- If the total number of items n in the data is
**even**, then

- If the number of items n in the data is
- Cumulative frequency of a particular value of the variable (or class) is the sum total of all the frequencies up to that value (or the class).
- There are two types of cumulative frequency distributions.
- cumulative frequency distribution of less than type.
- cumulative frequency distribution of more than type.

**Median of grouped data with class-intervals**

In this case, we first find the half of the total frequencies, i.e., n/2. The class in which n/2 lies is called the**median class**and the median lies in this class.

We use the following formula for finding the median.

- Where,

*l*= lower limit of the median class,

*n*= number of observations,

*cf*= cumulative frequency of the class preceding the median class,

*f*= frequency of the median class,

*h*= class size.

- Where,
- The three measures mean, mode and median are connected by the following relations.

**Mode = 3 median – 2 mean** - The graphical representation of a cumulative frequency distribution is called an ogive or cumulative frequency curve.
- We can draw two types of ogives for a frequency distribution. These are less than ogive and more than ogive.
- For less than ogive, we plot the points corresponding to the ordered pairs given by (upper limit, corresponding less than cumulative frequency). After joining these points by a free hand curve, we get an ogive of less than type.
- For more than ogive, we plot the points corresponding to the ordered pairs given by (lower limit, corresponding more than cumulative frequency). After joining these points by a free hand curve, we get an ogive of more than type.
- Ogive can be used to estimate the median of data. There are two methods to do so.
**First method**: Mark a point corresponding to n/2, where n is the total frequency, on cumulative frequency axis (y-axis).

From this point, draw a line parallel to x-axis to cut the ogive at a point.

From this point, draw a line perpendicular to the x-axis to get another point.

The abscissa of this point gives median.**Second method**: Draw both the ogives (less than ogive and more than ogive) on the same graph paper which cut each other at a point.

From this point, draw a line perpendicular to the x-axis, to get another point.

The abscissa of this point gives median.